The Mass of Composite Systems
The rest mass of a composite system is not the sum of the rest masses of the parts, unless all the parts are at rest. The total mass of a composite system includes the kinetic energy and field energy in the system.
The total energy E of a composite system can be determined by adding together the sum of the energies of its components. The total momentum of the system, a vector quantity, can also be computed by adding together the momenta of all its components. Given the total energy E and the length (magnitude) p of the total momentum vector, the invariant mass is given by:
In a mathematical system where c = 1, for systems of particles (whether bound or unbound) the total system invariant mass is given equivalently by the following:
Where, again, the particle momenta are first summed as vectors, and then the square of their resulting total magnitude (Euclidean norm) is used. This results in a scalar number, which is subtracted from the scalar value of the square of the total energy.
For such a system, in the special center of momentum frame where momenta sum to zero, again the system mass (called the invariant mass) corresponds to the total system energy or, in units where c=1, is identical to it. This invariant mass for a system remains the same quantity in any inertial frame, although the system total energy and total momenta are functions of the particular inertial frame which is chosen, and will vary in such a way between inertial frames as to keep the invariant mass the same for all observers. Invariant mass thus functions for systems of particles in the same capacity as "rest mass" does for single particles.
Note that the invariant mass of an isolated system (i.e., one closed to both mass and energy) is also independent of observer or inertial frame, and is a constant, conserved quantity for isolated systems and single observers, even during chemical and nuclear reactions. The concept of invariant mass is widely used in particle physics, because the invariant mass of a particle's decay products is equal to its rest mass. This is used to make measurements of the mass of particles like the Z boson or the top quark.
Read more about this topic: Mass In Special Relativity
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